One method disclosed herein includes forming a layer of silicon/germanium having a germanium concentration of at least 30% on a semiconducting substrate, forming a plurality of spaced-apart trenches that extend through the layer of silicon/germanium and at least partially into the semiconducting substrate, wherein the trenches define a fin structure for the device comprised of a portion of the substrate and a portion of the layer of silicon/germanium, the portion of the layer of silicon/germanium having a first cross-sectional configuration, forming a layer of insulating material in the trenches and above the fin structure, performing an anneal process on the device so as to cause the first cross-sectional configuration of the layer of silicon/germanium to change to a second cross-sectional configuration that is different from the first cross-sectional configuration, and forming a final gate structure around at least a portion of the layer of silicon/germanium having the second cross-sectional configuration.